A187789 a(n) is the start position for a Sankt-Petrus-game with n white and n black stones and the least step A187788(n).

2, 3, 2, 8, 5, 5, 2, 7, 1, 17, 15, 4, 8, 1, 2, 30, 26, 11, 35, 7, 26, 27, 23, 44, 24, 30, 6, 39, 53, 18, 2, 15, 61, 40, 30, 68, 44, 32, 78, 29, 81, 15, 19, 76, 51, 67, 40, 19, 53, 42, 53, 3, 74, 103, 73, 35, 105, 78, 110, 105, 76, 61, 2, 5, 48, 128, 82, 36, 37, 63, 88, 87, 31, 123, 93, 126, 2, 1, 156, 89, 33, 160, 90, 135, 124, 136, 145, 79, 42, 26, 104, 94, 67, 44, 186, 30, 133, 137, 40, 118

Offset

1,1

Comments

Beginning at the position a(n) with the least step A187788(n) (n-1) white stones were eliminated; then starting at the position 1 of the last white stone, n black stones were eliminated.

References

W. Ahrens, Das Josephusspiel, Archiv für Kulturgeschichte, Jg 11(1913), 129-151.

Links

Table of n, a(n) for n=1..100.

R. Baumann, Das Josephus-Problem, LOG IN, Heft Nr. 165, pp. 68-71, 2010 (in German).

Index entries for sequences related to the Josephus Problem

Example

n=8; WWBBWBBWWBBWWBWB; step=A187788(8)=3; start=a(8)=7; elimination: white stones: {9,12,15,2,5,8,13}, black stones: {4,10,16,6,14,7,3,11}.

Maple

stone:=proc(n1)
local n, j, k, h, z, zp: global a, m, s:
n:=2*n1: m:=m+1:
for j from 1 to n-1 do z[j]:=z[j]+1: end do:
z[n]:=1: zp:=1:
for j from 1 to n1 do
for k from 1 to (s-2) do  zp:=z[zp]: end do:
  h:=z[zp]: z[zp]:=z[z[zp]]: zp:=z[zp]:
end do:
if (h=1) then a[n1]:=1: else a[n1]:=n+2-h: end if:
end proc:
m:=0: s:=1:
while (m < 100) do
s1:=s: s:=s+1: c:=1:
for p from 2 to 100 by 2 do  p1:=p-1: p2:=p+1:
  b:=(c+s1) mod p +1:
  if (b=1) and (a[p1]=0) then stone(p1): end if:
  c:=(b+s1) mod p2 +1:
  if (c=1) and (a[p]=0) then stone(p): end if:
end do:
end do:

Crossrefs

Cf. A187788, A206602.

Sequence in context: A096488 A011280 A136193 * A245922 A098513 A134347

Adjacent sequences: A187786 A187787 A187788 * A187790 A187791 A187792

Keyword

nonn

Author

Paul Weisenhorn, Jan 06 2013

Status

approved